3.1 Introduction
This chapter27examines a stochastic model of a nancial market with long-lived dividend-paying assets and endogenous market clearing asset prices. This model is a version of that proposed in Evstigneev et al.(2006), and then analyzed primarily in the context of evolutionary nance (for a survey of the eld see Evstigneev et al. (2009)). The main focus in evolutionary nance is on questions of "survival and extinction" of investment strategies (portfolio rules). In this chapter we analyze the model from a different perspective and treat its decision-theoretic framework as a game in which the payoffs of the players (investors) are de ned in terms of the growth rates of their relative wealth. We show that in the game under consideration the Kelly (1956) portfolio rule of "betting your beliefs" forms with probability one a unique symmetric Nash equilibrium strategy.
Game-theoretic models of asset markets dealing with relative wealth of in- vestors have been put forth by Bell and Cover (1980,1988). In the one-shot, two- person zero-sum models, each investor wishes to outperform any other investor. The solution concept used in their models is a Nash equilibrium de ned in terms 27 This chapter is based on the material of the paper by W. Bahsoun, I. Evstigneev and L. Xu "Almost
sure Nash equilibrium strategies in evolutionary models of asset markets," Working paper No 10-08 of the Mathematics Department of the University of Loughborough, February 2010.
of the expectations of random payoff functions. It is shown that anyone who devi- ates the log-optimal strategy results in a fall in the expected payoff. In this chapter, we consider a different (stronger) solution concept: almost sure Nash equilibrium. Any unilateral deviation from the Kelly rule leads to a decrease in the random payoff with probability one.
Another work which can be linked to this chapter is the paper of Alós-Ferrer and a Ania (2004). Both of us employ a game-theoretic asset market model and focus on the performance of investment strategies. But their model is limited in nitely many states of the world and allows for redundant assets. Whilst our model deals with in nitely states of the world and does not allow for redundant assets. Another difference between our works lies in the solution concept. They de ne a pure- strategy Nash equilibrium in terms of expected payoff as a game solution concept, but we employ a (stronger) solution concept—almost sure Nash equilibrium with respect to the random payoff.
The present chapter focuses on optimality almost surely, which is character- istic for capital growth theory (Kelly (1956), Breiman (1961), Algoet and Cover (1988), Hakansson and Ziemba (1995), Maclean et al. (2010)). Most of the previ- ous research deals with asset market models with exogenous asset prices. Results related to evolutionary nance may be regarded as analogues, and in certain cases as generalizations, of those pertaining to classical models of capital growth. The main difference between the two modelling frameworks lies in the fact that in
the former the accumulation of wealth of each investor might depend (via the en- dogenous price formation mechanism) not only on his/her strategy, but also on the strategies used by the other investors. Therefore in the present context a game- theoretic model, rather than a single-agent optimization framework, is a suitable setting for the analysis of questions related to capital growth.
This chapter is organized as follows. Section 3.2 is the model description, Sec- tion 3.3 states the main results and Section 3.4 provides the proof of the main theorem. The Appendix 3.5 contains the proof of a technical lemma.
3.2 The model
We consider a general asset market with K 2assets and N 2investors (traders) acting in the market. The market is in uenced by random factors mod- elled in terms of independent identically distributed random elements s1; s2; :::
in a measurable space S. At each trading date t = 1; 2; ::: one unit of asset k = 1; 2; :::; K yields nonnegative dividends Dk(st) 0depending on the “state
of the world” stat date t. The dividends Dk(st)are measurable and satisfy K
X
k=1
Dk(s) > 0 for all s: (3.1)
This condition means that in each random situation at least one asset yields a strictly positive dividend. The total volume (the number of units) of asset k traded in the market at date t is
Vt;k = Vt;k(st) > 0;
where st := (s
t = 0, Vt;kis a constant number, and for t 1, Vt;k(st)is a measurable function of
st.
Denote by pt 2 RK+ the vector of market prices of the assets. For each k =
1; :::; K, the coordinate pt;k of pt = (pt;1; :::; pt;K)stands for the price of one unit
of asset k at date t. A portfolio of investor i at date t = 0; 1; ::: is speci ed by a vector xi
t = (xit;1; :::; xit;K) 2 RK+ where xit;k is the amount (the number of units)
of asset k. It means the numbers of units of all assets purchased by investor i. The scalar product hpt; xiti =
PK
k=1pt;kx i
t;kexpresses the value of the investor i's
portfolio xi
tat date t in terms of the prices pt;k.
At date t = 0 investor i has initial endowments wi
0 > 0that can be viewed as
his/her budget at date 0. Investor i's budget (wealth) at date t 1consists of two components: the dividends hDt; xit 1i paid by the portfolio xit 1 and the market
value hpt; xit 1i of the portfolio xit 1expressed in terms of the today's prices pt
wti :=hDt+ pt; xit 1i; (3.2)
where
Dt:= D(st) := (D1(st); :::; DK(st)):
Let t= t(st 1)be a fraction of the budget invested into assets. Suppose that the
investment rate 0 < t(st 1) < 1is the same for all the traders, although in general
it may depend on time and random factors. We assume that t is predictable: it
depends on the history st 1of the process (s
1 t can represent a fraction of money used for supporting investors's life or
business, which can also be understood as the tax rate or the consumption rate. The assumption that 1 t is the same for all the investors is quite natural in the
former case. In the latter case it is indispensable since we focus in this work on the analysis of the comparative performance of trading strategies (portfolio rules) in the long run. Without this assumption, an analysis of this kind does not make sense: a seemingly worse performance of a portfolio rule in the long run might be simply due to a higher consumption rate of the investor.
Suppose that the function t(st 1)is measurable (for t = 0; 1 it is constant)
and not greater than the supply growth rate of each asset k, i.e., that satis es the following condition
t(st 1) < Vt;k(st)=Vt 1;k(st 1): (3.3)
This condition holds, in particular, when the total mass Vt;k(st) of each asset k
does not decrease, i.e., when the right-hand side of (3.3) is not less than one. But (3.3) does not exclude the situation when Vt;k decreases at some rate, not faster
than t.
An investment strategy (portfolio rule) is speci ed by a vector of investment proportions i
t = ( i 1; :::;
i
K)2 K according to which he/she plans to distribute
the available budget between assets at each date t. Vector i
simplex
K
:=f(a1; :::; aK) 0 : a1+ ::: + aK = 1g:
Strategies of this kind are called xed-mix, or constant proportions, portfolio rules: they prescribe to select investment proportions at time 0 and keep them xed over the whole in nite time horizon.
The class of xed-mix investment strategies is quite widely used in nancial theory and practice, playing an important role in portfolio theory; see, e.g., Per- old and Sharpe (1988) and Browne (1998). Investors continuously rebalance their portfolios in order to keep xed constant investment proportions. Under certain conditions, strategies with constant investment proportions lead to the growth of the portfolio value (“volatility pumping”— Luenberger (1998)). From the theo- retical standpoint, this class of portfolio rules provides a convenient laboratory for the analysis of questions we are interested in. It makes it possible to formalize in a clear and compact way the concept of the type of an investor which determines the performance of his/her portfolio rule in the long run.
In the model at hand, the asset market evolves in time, remaining in the state of a dynamic short-run (temporary) equilibrium28. The notion of market equilibrium is de ned as follows. Suppose each investor i = 1; :::; N has selected a strategy— a vector of investment proportions i
= ( i1; :::; iK) 2 K at each date t. Then
28 In this paper, we use the term “equilibrium” in two different meanings. Here it is related to market
equilibrium: a situation when asset supply is equal to asset demand. Later, the same term will appear in a game-theoretic context. It will pertain to Nash equilibrium strategies in a certain dynamic game.
at date t 0, the amount of wealth allocated to asset k by trader i is t ikwit.
By summing up all traders wealth invested in asset k, we have the total amount demand of asset k, tPNi=1 ikwit, where wit is investor i's budget at time t. At
each trading date t, the market satis es the market clearing condition: asset supply is equal to asset demand, making it possible to determine the equilibrium price pt;k
of each asset k from the following equations pt;kVt;k = t N X i=1 i kwit; k = 1; :::; K: (3.4)
The left-hand side of (3.4) is the total market value of the supply of asset k at date t (recall that the amount of each asset k at date t is Vt;k). The right-hand side
represents the total wealth invested in asset k by all the investors. Equilibrium implies the equality in (3.4). The portfolios xi
t = (xit;1; :::; xit;K)are determined
by the investment proportions i 1; :::;
i
K chosen by the traders at time t by the
formulas xit;k = t i kwti pt;k ; k = 1; :::; K; i = 1; :::; N: (3.5) Note that for t 1, investor i's budget can be expressed by (3.2) and the price vector ptis determined implicitly as the solution to the system of equations (3.4),
which can be written as pt;kVt;k = t N X i=1 i khDt+ pt; xit 1i; k = 1; :::; K: (3.6)
It can be shown that under assumption (3.3) a non-negative vector pt satisfying
these equations exists and is unique (for any stand any feasible xi t 1and
Proposition 2.1).
Equations (3.5) make sense only if pt;k > 0, or equivalently, if the aggregate
demand for each asset (under the equilibrium prices) is strictly positive. We say a strategy pro le ( 1; :::; N)is admissible if it guarantees that each asset k has a
strictly positive equilibrium price through the above recursive procedure described step by step from (3.4)–(3.5). In what follows, we will deal only with such strategy pro les so as to guarantee the random dynamical system under consideration is well-de ned.
A path of market dynamics can be generated recursively according to a strategy pro le ( 1; :::; N)of all the investors and their initial wealth w1
0; :::; wN0
(pt; x1t; :::; x N
t ); (3.7)
where pt = pt(st) is the price vectors and xit = xit(st) is investor i's portfolio
(i.e., the amounts of units of all assets purchased by investor i). Since the hypoth- esis of admissibility is throughout this chapter, we obtain that all the portfolios xi
t = (xit;1; :::; xit;K)are non-zero and the wealth wti = hDt+ pt; xit 1i of each in-
vestor is strictly positive from the above equilibrium path. Further, by summing up equations (3.5) over i = 1; :::; N, we nd that
N X i=1 xit;k = PN i=1 t ikwti pt;k = pt;kVt;k pt;k = Vt;k (3.8)
(the market clears) for every asset k and each date t 1. Thus for every equilib- rium state of the market (pt; x1t; :::; xNt ), we have pt > 0, xit6= 0 and (3.8).
We shall provide a simple suf cient condition for making a strategy pro le admissible which can guarantee asset prices are strictly positive and market dy- namics are well de ned. Suppose that some trader, say trader 1, selects a com- pletely mixed portfolio rule to invest into all the assets, i.e., each coordinate of the investment strategy vector of trader 1 at time t = 0; 1; ::: must be strictly posi- tive. Then a strategy pro le containing this portfolio rule is admissible. Indeed, for t = 0, we get from (3.4) that p0;k 0V0;k1
1
kw10 > 0 and from (3.5) that
x10 = (x10;1; :::; x10;K) > 0(coordinatewise). Assuming that x1t 1 > 0and arguing by induction, we obtain hDt+ pt; x1t 1i hDt; x1t 1i > 0 in view of (3.1), which
in turn yields pt> 0and x1t > 0by virtue of (3.4) and (3.5), as long as 1 k > 0.
3.3 The main results Let ( 1
; :::; N)be an admissible strategy pro le of the investors. Given this strategy pro le and initial endowments, a path of market dynamics (3.7) can be generated in accordance with the equations (3.4)–(3.5). As above, let wi
t denote
the investor i's wealth available at date t 0. If t = 0, then the initial endowment wi
0 of investor i is assumed as a constant number. If t 1, then wit = wit(st)is
de ned as a measurable function of stgiven by formula (3.2). As we have noted
above, wi
t(st) > 0.
the investors. The relative wealth of investor i; i = 1; 2; :::N are given by rit= w i t Wt ;
where Wt :=PNi=1witis the total market wealth. Given a strategy pro le (
1; :::; N)
the performance of a strategy iused by investor i can be characterized by the ra-
tio between investor i's relative wealth and the coalition fj : j 6= ig of i's rivals in the game under consideration
i := lim supt!1 1 t ln rit P j6=ir j t : (3.9)
The random variable i
= i(s1; 1; :::; N)depends on the strategy pro le ( 1; :::; N) and on the whole history s1 := (s
1; s2; :::) of states of the world from time 1 to
1; playing the role of the (random) payoff function of player i. Further, this pay- off function re ects the fact that the performance of investor i is in uenced not only by his/her investment strategy, but also his/her rivals'.
De nition 3.1. We shall say that a strategy forms a symmetric Nash equilib- rium almost surely (a.s.) if
i(s1; ; :::; ) i(s1; ; :::; ; :::; )(a.s.) (3.10)
for every i, each strategy of investor i and each set of initial endowments w1 0 >
0; :::; wN
0 > 0. The Nash equilibrium is called strict if the inequality in (3.10) is
strict.
The strategy pro le ; :::; is admissible (recall that we consider only ad- missible strategy pro les) if and only if the vector is strictly positive. This
observation is immediate from (3.6).
Assume that the total mass Vt;kof each asset k grows (or decreases) at the same
rate t= t(st 1) > 0:
Vt;k=Vt 1;k = t(t 1): (3.11)
Thus
Vt;k(st 1) = t(st 1)::: 2(s1) 1Vk; (3.12)
where Vk > 0(k = 1; 2; :::; K) are the initial amounts of the assets. The growth
rate process t(like the investment rate process t) is predictable: tdepends only
on the history st 1of the states of the world up to time t 1. In the case of dividend-
paying assets involving investments in the real economy, assumption (3.11) means that the economic system under consideration is on a balanced growth path.
De ne the relative dividends of the assets k = 1; :::; K by Rk(st) =
Dk(st)Vk
PK
m=1Dm(st)Vm
(3.13) It follows from (3.12) that
Rt;k =
Dt;kVt 1;k
PK
m=1Dt;mVt 1;m
:
where Rt;k = Rk(st)and Dt;k = Dk(st). Indeed, from (3.12), we have
Rk(st) = Dt;kVt 1;k PK m=1Dt;mVt 1;m = Dt;k t 1(s t 2)::: 2(s1) 1Vk PK m=1Dt;m t 1(st 2)::: 2(s1) 1Vm = PKDk(st)Vk m=1Dm(st)Vm :
De ne
k = ERk(st); k = 1; 2; :::; K (3.14)
and put = ( 1; :::; K). The investment strategy speci ed by (3.14) may be regarded as a generalization of the Kelly portfolio rule of “betting your beliefs”. In this context it takes on the form of a rational expectations strategy (see Chap- ter 2). It is expressed in terms of the expected relative dividends, according to which investors distribute wealth across assets in accordance with the proportions of the expected relative dividends (which do not depend on t because the random elements stare i.i.d.).
Assume that the following conditions hold.
(R1) For each k, the expectation ERk(st)is strictly positive.
(R2) The functions R1(s); :::; RK(s) are linearly independent with respect to
the probability distribution of st, i.e., the equality
P
kRk(st) = 0 holding a.s.
for some constants k implies that 1 = ::: = K = 0.
(R3) There exist constants 0 < 0 < 00 < 1such that the process t(s t 1) := t(st 1)= t(s t 1) satis es 0 t(st 1) 00.
Condition (R1) implies that the vector has strictly positive coordinates. Hy- pothesis (R2) can be interpreted as the absence of redundant assets. Condition (R3) states that the discount factor tcannot be too close to 0 and 1. Under these
assumptions, the following theorem holds.
Theorem 3.1 The portfolio rule is a unique strategy forming a symmetric Nash equilibrium a.s.. This equilibrium is strict.
This result implies the following property of the portfolio rule . If all the investors except one, say investor i, use the strategy and i uses any other strategy
distinct from , then the relative wealth ri t=
P
j6=ir j
t of i tends to zero at the
exponential rate i
< 0(a.s.). In other words, the coalition of the Kelly investors drives the non-Kelly one out of the market. We note that this result (without an exponential estimate of the convergence rate) can be derived from Theorem 1 in Evstigneev et al. (2008) under the assumptions that the state space S is nite and all the strategies under consideration are completely mixed.
3.4 Proofs
For the proof of Theorem 3.1 we begin with a system of equations governing the dynamics of the market shares ri
t := wit=
P
jw j
t. Consider the path (3.7) of the
random dynamical system generated by ( 1; :::; N) and the sequence of vectors
rt = (rt1; :::; rtN)of the market shares of the investors at date t. Proposition 2.2
states the following equations hold rit+1= K X k=1 [ t+1h k; rt+1i + (1 t+1)Rt+1;k] i krit h k; rti ; (3.15)
i = 1; :::; N, t 0. From the above equations we observe that the market share of investor i evolves in terms of the interaction of the strategies 1
; :::; N.
investors. The above system can be reduced to the case of two investors. And the ratio of the market shares of investors 1 and 2 at date t is
zt := rt1=r 2 t = w 1 t=w 2 t;
where investor 1 and investor 2 select the Kelly rule = ( 1; :::; K)and = ( 1; :::; K);respectively. The dynamics of ztare described by the following equa-
tion zt+1 = zt PK k=1[ t+1 k + (1 t+1)Rt+1;k] k kzt+ k PK k=1[ t+1 k + (1 t+1)Rt+1;k] k kzt+ k : (3.16)
For a proof of (3.16), see the proof of proposition 2.4 in Appendix 2.5.
Consider any measurable relative dividend vector function R(s) = (R1(s); :::; RK(s))
on S satisfying (R1) and (R2). For any = ( 1; :::; K) 2 K, 0 00(see
(R3)) and 2 (0; 1], de ne F ( ; ; s) := PK k=1[ k+ (1 )Rk(s)] k k + k(1 ) PK k=1[ k+ (1 )Rk(s)] k k + k(1 ) ; (3.17)
where k = ERk(s) (E( ) is the unconditional expectation with respect to the
given probability P on S). The function F ( ; ; s) is well-de ned and takes on nite strictly positive values.
Lemma 3.1 For any 2 K distinct from there exist constants H > 0 and
> 0such that
E minfH; ln F ( ; ; s)g (3.18)
for all 2 (0; 1] and all 2 [ 0; 00].
lemma is routine, but rather lengthy, and we relegate it to the Appendix 3.5. Proof of Theorem 3.1. To demonstrate that forms a strict symmetric Nash equilibrium a.s. it is suf cient to consider the case of two investors 1 and 2, using
and , and show that
lim inft!1 1
t ln zt> 0(a.s.); (3.19)
where ztis the ratio of the market shares of 1 and 2.
To prove that the problem reduces to the case N = 2, let us rst observe that by virtue of symmetry, it is suf cient to verify the property (3.10) for i = N. Suppose investors i = 1; 2; :::; N 1use and investor N uses 6= . Then the total market share rt := r1 t + ::: + r N 1 t of i = 1; 2; :::; N 1satis es rt+1= K X k=1 t+1( krt+1+ krNt+1) + (1 t+1)Rt+1;k k rt krt + krtN : (3.20) This relation is obtained by summing up equations (3.15) over i = 1; 2; :::; N 1. At the same time, by virtue of (3.15), we have
rt+1N = K X k=1 [ t+1( krt+1+ krNt+1) + (1 t+1)Rt+1;k] krtN krt + krNt : (3.21) Thus the vector (rt; rN
t )evolves in time as the vector (~rt1; ~r2t)of market shares of
two investors using the strategies and , respectively. If (3.19) holds, then
N( ; :::; ; ) = lim sup t!1 1 t ln wN t wt = lim supt!1 1 t ln rNt rt = lim supt!1 1 t ln ~ r2t ~ r1 t = lim supt!1 1 t ln zt = lim inft!1 1 t ln zt < 0 = N ( ; :::; ; )(a.s.); where the last equality holds because the market shares of all the investors remain
constant, as long as all of them use the same strategy. Indeed, if all N investors use the same strategy, say , ri
t; i = 1; 2; :::; N has the following equation from
(3.15) rit+1 = K X k=1 [ t+1h k; rt+1i + (1 t+1)Rt+1;k] k ri t h k; rti = rit K X k=1 [ t+1h k; rt+1i + (1 t+1)Rt+1;k] k k PN i=1r i t = rit K X k=1 [ t+1 k+ (1 t+1)Rt+1;k] = rit= ::: = ri0:
Let us verify (3.19). Put Gt = ln(zt=zt 1). Then T X t=1 Gt = T X t=1 (ln zt ln zt 1) = ln zT ln z0:
Therefore it suf ces to prove that lim infT !1T 1PTt=1Gt > 0a.s.: For any con-
stant H de ne GH
t := minfGt; Hg. Since GHt Gtit is suf cient to prove that
lim inf T !1 1 T T X t=1 GHt > 0(a.s.) (3.22) for some H. Observe that Gt+1= ln zt+1 zt = ln PK k=1[ t+1 k+ (1 t+1)Rt+1;k] k kzt+ k PK k=1[ t+1 k+ (1 t+1)Rt+1;k] k kzt+ k = ln PK k=1[ t+1 k+ (1 t+1)Rk(st+1)] k kr1t+ k(1 rt1) PK k=1[ t+1 k+ (1 t+1)Rk(st+1)] k kr1t+ k(1 rt1) = ln F t+1( ; r 1 t; st+1); (3.23) where r1
By virtue of Lemma 3.1, there exist H > 0 and > 0 such that EtGHt+1 ,
where Et( ) = E(jst)is the conditional expectation given stand
GHt+1(st+1) = minfH; ln F t+1(st)( ; r 1 t(s t); s t+1)g:
When computing EtGHt+1we x stand take the unconditional expectation of GHt+1
with respect to st+1, which is justi ed because stand st+1 are independent.
Finally, we have 1 T T X t=1 GHt = 1 T T X t=1 Et 1GHt + 1 T T X t=1 (GHt Et 1GHt ): Since GH
t is uniformly bounded, we can apply to the process BtH := GHt Et 1GHt
the strong law of large numbers for martingale differences (see, e.g., Hall and Heyde (1980)), which yields 1
T
PT t=1B
H
t ! 0 (a.s.). Therefore lim inf T 1
PT t=1G
H t
, which proves (3.22).
Suppose a strategy 6= forms a symmetric Nash equilibrium with probabil- ity one. Then
0 = N(s1; ; :::; ) N(s1; ; :::; ; )(a.s.); (3.24) where N(s1; ; :::; ; ) = lim sup t!1 1 t ln rN t 1 rN t :
By interchanging and in formulas (3.20) and (3.21), we obtain that the vector (r1
t + ::: + r N 1
t ; rtN)evolves in time as the vector (^rt1; ^r2t)of market shares of two
this implies lim inft!11 t ln rN t 1 rN t > 0(a.s.).
Therefore N(s1; ; :::; ; ) > 0(a.s.), because lim sup lim inf, which yields
the inequality "<" in (3.24). This is a contradiction.
3.5 Appendix
Proof of Lemma 3.1. We observe that the function F ( ; ; s) is bounded below
F ( ; ; s) c2; (3.25)
where c := mink k(> 0):Indeed, F ( ; ; s) = A=B, where
A = K X k=1 [ k+ (1 )Rk(s)] k k + k(1 ) ; B = K X k=1 [ k+ (1 )Rk(s)] k k + k(1 ) : We observe A K X k=1 [ k+ (1 )Rk(s)] min k k k + k(1 ) min k k k + k(1 ) mink k maxk[ k + k(1 )] min k k; since max[ k + k(1 )] 1:
And B is bounded. Indeed, we have the following inequalities B K X k=1 k k k + k(1 ) 0c K X k=1 k 0c; and B max k k k + k(1 ) max k 1 k + (1 ) 1 mink k = 1 c; (3.26) since 0 k 1;and k + (1 ) k:
set K := fk : k = 0g is not empty. Then A = (1 )1 X k2K Rk(s) + X k =2K [ k+ (1 )Rk(s)] k k + k(1 ) and B = K X k =2K [ k+ (1 )Rk(s)] k k + k(1 ) :
By virtue of (R1), there exists > 0 such thatPk2KRk(s) for all s in a set S
with P (S) > 0. We have (1 00) 1S(s) (1 )1 X k2K Rk(s) A max k =2K k k + k(1 ) + (1 )1 1 mink =2K k 1 + 1: Therefore d1 1S(s) A D1;
where 1S(s) is the indicator function of the set S, d1 := (1 00) and D1 :=
1 + (mink =2K k) 1.
Also, we can see B is bounded. Indeed, we have